The Geometric Mean Value Of Product Of Non-zero Digits In Base 10 And The Mean Value Of Some Number Functions For Square Complements

This thesis mainly discusses the properties of some number theoretical functions. There are two chapters in this thesis. In the first chapter, we studied the geometric mean value of product of non-zero digits in base 10.For every positive integer N , letwhere We define the product function of non-zero digits in base 10 of n as follows:and we define the geometric mean value of product of non-zero digits in base 10 as follows:In this chapter, we studied the arithmetical property of A(N) and gave an exact calculating formula for it.In the second chapter, we studied the mean value of some number functions for square complements. For a positive integer n , we have n =u2v, where u is a positive integer, v has no square factor. c(n) be the squarecomplements of n , d(n) and cr(n) be the divisor function s (n)when s = 0 and s = 1.In this chapter, we using analytic method studied the asymptoticproperties of the mean value d(c(n)) and (c(n)), and obtained two sharpasymptotic formulas.